Boosting the Model Discovery of Hybrid Dynamical Systems in An Informed Sparse Regression Approach

We present an efficient data-driven sparse identification of dynamical systems. The work aims at reconstructing the different sets of governing equations and identifying discontinuity surfaces in hybrid systems when the number of discontinuities is known a priori. In a two-stages approach, we first locate the switches between separate vector fields. Then, the dynamics among the manifolds are regressed, in this case by making use of the existing algorithm of Brunton et al. [1]. The reconstruction of the discontinuity surfaces comes as the outcome of a statistical analysis implemented via symbolic regression with small clusters (micro-clusters) and a rigid library of models. These allow to classify all the feasible discontinuities that are clustered and to reduce them into the actual discontinuity surfaces. The performances of the sparse regression hybrid model discovery are tested on two numerical examples, namely, a canonical spring-mass hopper and a free/impact electromagnetic energy harvester, engineering archetypes characterized by the presence of a single and double discontinuity, respectively. Results show that a supervised approach, i.e. where the number of discontinuities is preassigned, is computationally efficient and it determines accurately both discontinuities and set of governing equations. A large improvement in the time of computation is found with the maximum achievable reliability. Informed regression-based identification offers the prospect to outperform existing data-driven identification approaches for hybrid systems at the expense of instructing the algorithm for expected discontinuities.

Spherical data validation of rock discontinuities orientation from Drone-derived 3D Point Clouds

This work presents a methodology for the statistical validation of discontinuity surfaces obtained from point clouds using digital photogrammetry from drones. Our methodology allows you to review the quality of the data obtained with photogrammetry and decide whether these measurements are representative of the discontinuity surfaces that they analyze. It consists of three steps, the first one being a shape analysis that allows defining which statistical model should be used: Fisher for circularly symmetric clusters or Bingham fits better for axially symmetric clusters. This step also makes the most significant difference to other works since our methodology starts from the premise that not all discontinuity surfaces are flat. Therefore, Fisher parameters do not allow validating data that do not correspond to a plane. In the second step of the methodology, we calculate the consistency parameters that depend on the statistical model defined in step 1. The parameters are similar for both models; both estimate κ which indicates how much the sample is concentrated around the mean orientation and validates the existence of this and which is the value of the generating angle of a cone with a 95 % confidence limit that it contains within the mean orientation. Finally, step 3 is used when there are control measurements to compare the point cloud data and define if both samples characterize the same discontinuity surface in the rock mass. The results obtained on a rock outcrop allowed us to observe that the measurements obtained from the drone faithfully represent the discontinuity surface analyzed when these were compared with the measurements made manually with the compass. Furthermore, the dispersion parameters (

Boosting the Model Discovery of Hybrid Dynamical Systems in an Informed Sparse Regression Approach

We present an efficient data-driven sparse identification of dynamical systems. The work aims at reconstructing the different sets of governing equations and identify discontinuity surfaces in hybrid systems when the number of discontinuities is known a priori. In our approach, we first focus to identify the switches between the separate vector fields. Then, the dynamics among the manifolds are regressed by making use of the model discovery algorithm of Brunton et al. [1]. The reconstruction of the discontinuity surfaces comes as the outcome of a statistical analysis implemented via symbolic regression with small clusters (micro-clusters) and a rigid library of models. This allows to identify all the many possible switch points that are clustered to determine the actual discontinuity surfaces. The performances of the method are tested on two numerical examples, namely, a canonical spring–mass hopper and a free/impact electromagnetic energy harvester. These applications are characterized by the presence of a single and double discontinuity, respectively. The analyses demonstrate that in the supervised approach, i.e. where the number of discontinuities is preassigned, we are capable to determine accurately both discontinuities and set of governing equations. It is found a great improvement in time of computation reaching the maximum achievable reliability that outperform existing data-driven identification approaches for hybrid systems.

Optimal control of the radiation heat exchange equations for multi-component media

An analysis of optimal control problems for nonlinear elliptic equations modeling complex heat transfer with Fresnel conjugation conditions on the discontinuity surfaces of the refractive index is presented. Conditions for the solvability of extremal problems and the nondegeneracy of the optimality system are obtained. For the control problem with boundary observation, the bang-bang property is set.

Nonlinear Behavior of a Novel Switching Jerk System

This paper proposes a novel chaotic jerk system, which is defined on four domains, separated by codimension-2 discontinuity surfaces. The dynamics of the proposed system are conveniently described and analyzed through a generalization of the Poincaré map which is constructed via an explicit solution of each subsystem. This provides an approach to formulate a robust bifurcation problem as a nonlinear inhomogeneous eigenvalue problem. Also, we establish some criteria for the existence of a period-doubling bifurcation and discuss some of the interesting categories of complex behavior such as multiple period-doubling bifurcations and chaotic behavior when the trajectory undergoes a segment of sliding motion. Our results emphasize that the sharp switches in the behavior are mainly responsible for generating new and unique qualitative behavior of a simple linear system as compared to the nonlinear continuous system.